Detectors

Goal: Convert photons to an electronic signal (apologies to photography...)

with as little accompanying noise as possibleideally at the quantum limit enforced by the photons.

with as much conversion efficiency as possible1 photon yields 1 electron (or ideally a bunch of electrons)

Primary Detection Methods

Bulk thermal response (bolometry)incident radiation chages the temperature of the detectorelectrical resistance changes with temperature

Conversion of photons to ''free'' electronsquantum responsephotoelectric or solid state detection

Coherent detectionsense wave nature (phase) of the photons primarily through heterodyning to lower frequencies

   

Electron response -- free electrons / carriers

A free electron is a detectable electron

an electron can be free in space -- photoelectric effector it can be ''free'' within a crystal lattice -- solid state detection

Metals are characterized by a work function which determines the energy difference between the highest energy state for an electron within the metal and the energy of an electron in free space.

The Photoelectric Effect

A photon with energy in excess of this work function will liberate a free, detectable, electron -- the photoelectric effect Heated metals will emit free electrons -- those with thermal

energy in excess of the material's work function -- thermionic emission via a Boltzmann law.

   

Photomultipliers are based on the cascade amplification of individual electrons liberated by the photoelectric effect

Work functions for metals are typically a few electron volts

1 eV = 1240 nm

The Photoelectric Effect

http://laxmi.nuc.ucla.edu:8248/M248_99/autorad/Scint/pmt.htmll

   

Photomultipliers are based on the cascade amplification of individual electrons liberated by the photoelectric effect

Work functions for metals are typically a few electron volts

1 eV = 1240 nmPhotocathodes can be engineered to have sensitivity out to 1.5 um (obviously not using pure metals...)

The Photoelectric Effect

http://hyperphysics.phy-astr.gsu.edu/hbase/tables/photoelec.html

   

Shortcomings of photomultipliers

poor wavelength coverage (<1.5um) poor quantum efficiency (<20% conversion of photons to electrons) thermally emitted electrons -- particularly for long-wavelength devices. large single-detector area (overcome with devices like micro-channel plates)

One big advantage -- photon counting

The Photoelectric Effect

Modern detection systems use semiconductor detectors which mimic the photoelectric effect in the solid state.

Photons create “free” electrons within the confines of the crystal lattice.

Discussion requires an understanding of conductivity, or lack thereof, in crystalline materials.

   

Detectors

   

Electronic Energy Levels in SolidsElectrons are fermions and must obey exclusion rules.

At the atomic level, electrons fill available energy states up to the valence levels... 1s, 2s, 2p, 3s, 3p, 3d, ... with two spin states in each energy state.

In a solid, proximity leads to splitting of the degenerate valence energy levels due to interference with neighboring atoms.

Lower energy levels remain tightly bound to their atoms. Higher (possibly unoccupied) levels become states of the entire crystal structure.

   

Idealized Electrical ConductionIgnoring the lattice sites for the moment, “free” electrons in a metal (or semiconductor) are particles in a box.

Momentum increases linearly with quantum state, energy quadratically.

As particles are dropped into the box they fill the lowest energy states.

Although the mean energy of particles is large (~Ef) the net momentum is zero.

Application of an electrical potential can elevate electrons into higher energy states -- conduction.

   

Electrical Conduction

The small disequilibrium between the right and left halves of the parabola represents electronic conduction.

Most electrons in the material are still in momentum states which cancel each other.

Only a limited number of electrons at the top of the Fermi sea participate in conduction.

   

Electrical ConductionIn real materials the lattice sites produce a periodic potential. When the electron DeBroglie wavelength becomes resonant with the lattice spacing significant scattering occurs.

The dispersion relation exhibits breaks at these boundarys providing an alternative view of the nature of conduction “bands”.

These breaks are caused by the mutual interference of the wavefunctions being reflected at each potential boundary.

   

Electronic Energy Levels in SolidsElectrons in the higher unbound states form a degenerate Fermi gas.

The bulk of the lower unbound states are completely filled with electrons (imagine dropping fermions into the classical infinite square well).

Only the electrons at the top of the Fermi sea can hope to change their energy/momentum states

   

Electronic Energy Levels in ConductorsAn alternative approach:

At large separations, electronic orbitals have “atomic” characteristics.

As atomic separation decreases these degenerate states must split under the interacting potential of all of the nuclei in the crystal.

The ensemble of split energy levels is a “band” which may be

– full, partially filled and/or overlapping with other bands

Electrons which have immediately adjacent energy states can change state and thus “conduct”

   

Electronic Energy Levels in Conductors

Sodium, a metal, has a single 3s valence band electron.

the 3s state is only half filled since any atom can have a second electron in this state with opposite spin.

In crystalline sodium (recall any chunk of metal is an assemblage of crystalline domains) the 3s state is shared by the entire crystal.

There are 2N (twice the number of atoms) 3s states in the crystal and only N electrons.

Conduction is “easy” since a valence electron sees a variety of nearby open energy/momentum states.

Each atom brings a fixed number of states and a fixed number of electrons. It is natural that bands sometimes end up exactly full.

   

Electronic Energy Levels in ConductorsCopper has a single 4s electron making it a natural conductor.

It also has an electronic configuration where the 4s band overlaps the 3d band at the crystalline interatomic spacing.

Overlap between bands also provides access to infinitesimally different energy states, permitting conduction (e.g. Magnesium – which has a filled 3s state is still an electrical conductor).

Note that there are about 10²² atoms in a fist sized chunk of metal. Each atom contributes a couple of 4s energy states.The energy states span a couple of eV. The degenerate conduction band energy levels are about 10¯²² eV apart.

   

Electronic Energy Levels in InsulatorsInsulators have filled energy bands which do not overlap with adjacent energy bands for the interatomic equilibrium spacing.

There is no such thing as a “semiconductor”!

At T=0 a material will either have overlapping energy states and is a conductor, or it will have a “bandgap” above a completely filled energy state and be an insulator.

Semiconductivity (if that is a word) is a manifestation of Boltzmann factors at finite temperature – kT vs. the “bandgap”.

   

Band Filling and Band Interactions

   

http://pearl1.lanl.gov/periodic/default.htm

SemiconductorsSemiconductors can consist of pure elemental materials or alloys of different elements.

In either case, materials with complementary filled valence shells are likely semiconductors.

Carbon (diamond) is a “semiconductor” with an energy gap of 5.33 eV (0.23um)l

   

At T=0K, the world contains only conductors and insulators.

Above 0K, electrons at the top of the Fermi sea can be excited to higher energy states if the states are sufficiently ( ~kT ) close. Small bandgap materials are thus semiconductors with marginal electrical conductivity at room temperature due to thermally excited carriers.

The conductivity of metals improves at low temperatures. The conductivity of semiconductors declines.

Semiconductors

   

Semiconductor Conductivity

The population of electrons (holes) in the conduction (valence) band in an intrinsic semiconductor depends on temperature.

The electrons obey Fermi-Dirac statistics, but given the large size of the bandgap vs. kT, occupancy in the conduction band follows an apparent Boltzmann law.

E bandgap= 2∗E−E f

   

Room Temperature Electrical Conductivity in Crystalline Silicon

At 300K, kT = 0.026 eV ( 1 eV = 11,600K)

Silicon's bandgap is 1.11 eV

The electron effective mass is 1.1 m

– Electron density 1.3×1010 e cm−3

   

Room Temperature Electrical Conductivity in Crystalline Silicon

The resistance of a material is given by

– where l is the length traversed by the current and A is the resistor cross section. is the “conductivity” (which represents the current density induced by an electric field, E).

– Conductivity can be expressed in terms of the material carrier density if one knows the electron mobility, , which is the proportionality between carrier drift velocity and the electric field.

a one centimeter cubic piece of silicon should have a room temperature resistance of ...

At 77K this resistance falls by a factor of

1.3×1010 e cm−3

R =l

Aohm cm−1

= nq ⟨v⟩E

= nqJ = E = nq ⟨v⟩

e−1.1

2∗0.026 / e−1.1

2∗0.007 = e−21.1/e−82.4 = e−61.3 = 2.4×10−27